Here we recall that the function \(L(r)\)  is continuous and slowly varying at infinity.
Therefore, by the uniform convergence theorem ([1]})
or [2]},
there exist \(C_1>0\)  and \(C_2>0\) 
for any positive constants \(a\) , \(b\)  with \(0<a<b\) , such that
\(C_1\le \frac{L(\lambda r)}{L(r)}\le C_2, \quad r\ge 1, \ \lambda \in [a,b].\) 
